For non-uniformly expanding maps inducing w.r.t. a general
(i.e., not necessarily first) return time to Gibbs Markov maps, we provide
sufficient conditions for obtaining sharp estimates for the correlation function.
This applies to both the finite and the infinite measure setting. The
results are illustrated by non-Markov intervals maps with an indifferent
fixed point. This is joint work with Dalia Terhesiu.